Post Refinement Element Shape Improvement for Quadrilateral Meshes

Schneiders and Debye (1995) present two algorithms for quadrilateral mesh refinement. These algorithms refine quadrilateral meshes while maintaining mesh conformity. The first algorithm maintains conformity by introducing triangles. The second algorithm maintains conformity without triangles, but requires a larger degree of refinement. Both algorithms introduce nodes with non-optimal valences. Non-optimal valences create acute and obtuse angles, decreasing element quality. This paper presents techniques for improving the quality of quadrilateral meshes after Schneiders’ refinement. Improvement techniques use topology and node valence optimization rather than shape metrics; hence, improvement is computationally inexpensive. Meshes refined and subsequently topologically improved contain no triangles, even though triangles are initially introduced by Schneiders’ refinement. Triangle elimination is especially important for linear elements since linear triangles perform poorly. In addition, node valences are optimized, improving element quality. Introduction Mesh refinement is the process of taking an existing finite element mesh and changing the size, shape, and/or order of the elements in the mesh in order to increase the accuracy of the finite element solution. Much research has been done on mesh refinement. Rivara (1996) presents a method of refining triangular meshes. Since any arbitrary domain can be completely spanned with triangles, it is trivial to state that Rivara’s (1996) refinement results in meshes composed of entirely triangles. The same is not true, however, for quadrilateral meshes. Although refinement of an all quad mesh can result in another all quad mesh, precautions must be taken to ensure this is the case. In addition to ensuring that only quadrilaterals remain in the post refinement mesh, the quality of the quadrilaterals must also be considered. Schneiders and Debye (1995) present several algorithms for twodimensional quadrilateral refinement. These algorithms, however, maintain quads at the cost of introducing poor quality quads. On the other hand, one of the algorithms presented by Schneiders and Debye maintains high element quality by introducing triangles. This paper introduces an extension to Schneiders’ quadrilateral refinement in order to eliminate the introduction of triangles and poor quality quadrilaterals (see Figure 1a & b). Element quality is improved by optimization of local topology. Local topology quality can be measured by node valence. Node valence is defined as the number of edges attached to a node. Local topology is improved by eliminating high valence or low valence nodes. The optimal valence of a node in a quadrilateral mesh is four. Schneiders’ refinement leaves nodes with valences as high as eight and as low as three. It is impossible for all nodes in a mesh to have a valence of four and still provide transitioning between large and small elements. However, valences can be optimized ensuring that no nodes remain with a valence higher than five or lower than three and that the number of such nodes is minimized. The valence optimization presented is based on the cleanup operations presented by Zhu et al. (1991), Blacker and Stephenson (1990), Canann et. al. (1994), and Canann et al. (1996). Several extensions to those ideas are also presented, including new clean up patterns, experience with clean up patterns, diagonal swapping constraints, and cases where Laplacian smoothing poses problems after cleanup. (a) Schneiders’ 2-Refinement (b) 2-Refinement after (2-iter.) improvement and smoothing Figure 1 Schneiders’ 2-Refinement before and after cleanup Previous Research Schneiders and Debye (1995) present two algorithms for quadrilateral mesh refinement. The first is called 2-refinement because each edge in the refinement region is split into two smaller edges. The second is called 3-refinement because each edge in the refinement region is split into three smaller edges. In both algorithms, a series of nodes are marked where refinement is to be performed. Each quad which is adjacent to any node that was marked is replaced with a transition element template. The only difference between the implementation of 2-refinement and 3-refinement is which templates are used. Figure 1a illustrates 2-refinement. Numerous triangles have been introduces in the transition region between the refinement region and non-refinement regions. These triangles are undesirable, especially in linear analyses (Zienkiewicz, 1977) and should be eliminated if possible. Schneiders and Debye (1995) present a variation of 2-refinement which does not introduce triangles. This method works well when refining regular gridded quadrilateral meshes. However, this alternate method is not general enough to be applied to irregular quadrilateral meshes. In addition, this method requires direction information to be stored about the elements in the transition region. The need for this additional information, as well as its inability to be generalized for non-structured meshes, suggests that this is not the desired solution. Figure 2 illustrates 3-refinement. No triangles have been introduced in the transition region; however, the elements in the transition region are of marginal quality. Node valence can be used to measure element quality. In an optimal quadrilateral mesh, the majority of nodes have a valence of four. With four edges, each of the quadrilaterals adjacent to the node has, on average, an angle of 90o at the node. If a node has a valence greater than four, the average element angle at this node decreases. Likewise, as the valence decreases, the average element angle increases. Node A in Figure 2 has a valence of seven. Element 1 in Figure 2 has an angle of 90o at node A. However, the other six quadrilaterals using node A have an angle of 45o. This is far from optimal. Similarly, node B in Figure 2 has a valence of six. The quadrilaterals using node B have, on average, a face angle of 60o at node B. While this is better than at node A, it is still far from optimal. Much research with topological cleanup has been done. All cleanup operations can be described as combinations of five basic operations: element opens (Canann et. al., 1994), element closes (Zhu et al., 1991, Blacker and Stephenson 1990, and Canann et. al., 1994), diagonal swaps (Zhu et al., 1991, and Canann et. al., 1994), 2-edge node elimination (Zhu et al., 1991, and Canann et. al., 1994), and 2edge node insertion (Canann et. al., 1994). Figure 14 illustrates each of these operations.